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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
Practical Control System Design This book delivers real world experience covering full-scale industrial control design, for students and professional control engineers Inspired by the authors' industrial experience in control, Practical Control System Design: Real World Designs Implemented on Emulated Industrial Systems captures that experience, along with the necessary background theory, to enable readers to acquire the tools and skills necessary to tackle real world control engineering design problems. The book draws upon many industrial projects conducted by the authors and associates; these projects are used as case studies throughout the book, organized in the form of Virtual Laboratories so that readers can explore the studies at their own pace and to their own level of interest. The real-world designs include electromechanical servo systems, fluid storage, continuous steel casting, rolling mill center line gauge control, rocket dynamics and control, cross directional control in paper machines, audio quantisation, wind power generation (including 3 phase induction machines), and boiler control. To facilitate reader comprehension, the text is accompanied by software to access the individual experiments. A full Solutions Manual for the questions set in the text is available to instructors and practicing engineers. Background theory covered in the text includes control as an inverse problem, impact of disturbances and measurement noise, sensitivity functions, Laplace transforms, Z-Transforms, shift and delta operators, stability, PID design, time delay systems, periodic disturbances, Bode sensitivity trade-offs, state space models, linear quadratic regulators, Kalman filters, multivariable systems, anti-wind up strategies, Euler angles, rotational dynamics, conservation of mass, momentum and energy as well as control of non-linear systems. Practical Control System Design: Real World Designs Implemented on Emulated Industrial Systems is a highly practical reference on the subject, making it an ideal resource for undergraduate and graduate students on a range of control system design courses. The text also serves as an excellent refresher resource for engineers and practitioners.
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Table of Contents
Cover
Table of Contents
Title Page
Copyright
Preface
Note
About the Authors
Acknowledgements
About the Companion Website
Part I: Modelling and Analysis of Linear Systems
1 Introduction to Control System Design
1.1 Introduction
1.2 A Brief History of Control
1.3 Digital Control
1.4 Our Selection
1.5 Thinking Outside the Box
1.6 How the Book Is Organised
1.7 Testing the Reader's Understanding
1.8 Revision Questions
Further Reading
2 Control as an Inverse Problem
2.1 Introduction
2.2 The Elements
2.3 Using Eigenvalue Analysis
2.4 The Effect of Process and Disturbance Errors
2.5 Feedback Control
2.6 The Effect of Measurement Noise
2.7 Sensitivity Functions
2.8 Reducing the Impact of Disturbances and Model Error
2.9 Impact of Measurement Noise
2.10 Other Useful Sensitivity Functions
2.11 Stability (A First Look)
2.12 Sum of Sensitivity and Complementary Sensitivity
2.13 Revision Questions
Further Reading
3 Introduction to Modelling
3.1 Introduction
3.2 Physical Modelling
3.3 State‐Space Model Representation
3.4 Linearisation and Approximation
3.5 Revision Questions
Further Reading
4 Continuous‐Time Signals and Systems
4.1 Introduction
4.2 Linear Continuous‐Time Models
4.3 Laplace Transforms
4.4 Application of Laplace Transforms to Linear Differential Equations
4.5 A Heuristic Introduction to Laplace Transforms
4.6 Transfer Functions
4.7 Stability of Transfer Functions
4.8 Impulse Response of Continuous‐Time Linear Systems
4.9 Step Response
4.10 Steady‐State Response and Integral Action
4.11 Terms Used to Describe Step Responses
4.12 Frequency Response
4.13 Revision Questions
Further Reading
Notes
5 Laboratory 1: Modelling of an Electromechanical Servomechanism
5.1 Introduction
5.2 The Physical Apparatus
5.3 Estimation of Motor Parameters
5.4 Revision Questions
Further Reading
Notes
Part II: Control System Design Techniques for Linear Single‐Input Single‐Output Systems
6 Analysis of Linear Feedback Systems
6.1 Introduction
6.2 Feedback Structures
6.3 Nominal Sensitivity Functions
6.4 Analysing Stability Using the Characteristic Polynomial
6.5 Stability and Polynomial Analysis
6.6 Root Locus (RL)
6.7 Nominal Stability Using Frequency Response
6.8 Relative Stability: Stability Margins and Sensitivity Peaks
6.9 From Polar Plots to Bode Diagrams
6.10 Robustness
6.11 Revision Questions
Further Reading
Note
7 Design of Control Laws for Single‐Input Single‐Output Linear Systems
7.1 Introduction
7.2 Closed‐Loop Pole Assignment
7.3 Using Root Locus
7.4 All Stabilising Control Laws
7.5 Design Using the Youla–Kucera Parameterisation
7.6 Integral Action
7.7 Anti‐Windup
7.8 PID Design
7.9 Empirical Tuning
7.10 Ziegler–Nichols (Z–N) Oscillation Method
7.11 Two Degrees of Freedom Design
7.12 Disturbance Feedforward
7.13 Revision Questions
Further Reading
8 Laboratory 2: Position Control of Electromechanical Servomechanism
8.1 Introduction
8.2 Proportional Feedback
8.3 Using Proportional Plus Derivative Feedback
8.4 Tachometer Feedback
8.5 PID Design
8.6 Revision Questions
Further Reading
9 Laboratory 3: Continuous Casting Machine: Linear Considerations
9.1 Introduction
9.2 The Physical Equipment
9.3 Modelling of Continuous Casting Machine
9.4 Proportional Control
9.5 Response to Set‐Point Changes
9.6 Experiments
9.7 Effect of Measurement Noise
9.8 Pure Integral Control
9.9 PI Control
9.10 Feedforward Control
9.11 Revision Questions
Further Reading
10 Laboratory 4: Modelling and Control of Fluid Level in Tanks
10.1 Introduction
10.2 The Controllers
10.3 Physical Modelling
10.4 Closed‐Loop Level Control for a Single Tank
10.5 Closed‐Loop Level Control of Interconnected Tanks
10.6 Revision Questions
Further Reading
Notes
11 Laboratory 5: Wind Power (Mechanical Components)
11.1 Introduction
11.2 Yaw Control
11.3 Rotational Velocity Control
11.4 Pitch Control
11.5 Experiment: Testing the Pitch Controller
11.6 Revision Questions
Further Reading
Notes
Part III: More Complex Linear Single‐Input Single‐Output Systems
12 Time Delay Systems
12.1 Introduction
12.2 Transfer Function Analysis
12.3 Classical PID Design Revisited
12.4 Padé Approximation
12.5 Using the Youla–Kucera Parameterisation
12.6 Smith Predictor
12.7 Modern Interpretation of Smith Predictor
12.8 Sensitivity Trade‐Offs
12.9 Theoretical Analysis of Effect of Delay Errors on Smith Predictor
12.10 Revision Questions
Further Reading
13 Laboratory 6: Rolling Mill (Transport Delay)
13.1 Introduction
13.2 The Physical System
13.3 Modelling
13.4 Building a Model
13.5 Basic Control System Design
13.6 Linear Control Ignoring the Time Delay
13.7 Linear Control Based on Rational Approximation to the Time Delay
13.8 Control System Design Based on Smith Predictor
13.9 Use of a Soft Sensor
13.10 Robustness of BISRA Gauge
13.11 Revision Questions
Further Reading
Notes
14 Control System Design for Open‐Loop Unstable Systems
14.1 Introduction
14.2 Some Simple Examples of Open‐Loop Unstable Systems
14.3 All Stabilising Control Laws for Systems Having Undesirable Open‐Loop Poles
14.4 Revision Questions
Further Reading
Note
15 Laboratory 7: Control of a Rocket
15.1 Introduction
15.2 Dynamics of a Rocket in 2D Flight
15.3 Equilibrium
15.4 Linearised Model
15.5 Open‐Loop Flight
15.6 Controller Design for the Rocket
15.7 Experiment: Testing the Control Law
15.8 Revision Questions
Further Reading
Notes
16 Bode Sensitivity Trade‐Offs
16.1 Introduction
16.2 System Properties
16.3 Bode Integral Constraints
16.4 Examples of Bode Sensitivity Trade‐Offs
16.5 Bode Complementary Sensitivity Integrals
16.6 Bode Sensitivity for Time‐Delay Systems
16.7 Revision Questions
Further Reading
Note
Part IV: Sampled Data Control Systems
17 Principles of Sampled‐Data Control System Design
17.1 Introduction
17.2 A/D Conversion
17.3 Sampled Output Noise
17.4 D/A Conversion
17.5 Sampled‐Data Models
17.6 Shift Operator Models
17.7 Divided Difference Models
17.8 Euler Approximate Model
17.9 Euler Approximate Model in Delta Domain
17.10 Delta Analysis
17.11 Historical Notes
17.12 An Example of Shift and Delta Models
17.13 Sampled‐Data Stability
17.14 Bode Sensitivity Integrals (Sampled Data Case)
17.15 Sampling Zeros
17.16 Revision Questions
Further Reading
Notes
18 Laboratory 8: Audio Signal Processing and Optimal Noise Shaping Quantisers
18.1 Introduction
18.2 The Physical Apparatus
18.3 Psychoacoustic Issues
18.4 Nearest Neighbour Quantisation
18.5 Optimal Noise Shaping Quantiser
18.6 Utilising Your Own Hearing Sensitivity
18.7 Audio Quantisation from a Bode Sensitivity Integral Perspective
18.8 Audio Quantisation for More Complex Cases
18.9 Revision Questions
Further Reading
Part V: Simple Multivariable Control Problems
19 Tools Used for Simple Multivariable Control Problems
19.1 Introduction
19.2 Cascade Control
19.3 Imposed SISO Architectures
19.4 Relative Gain Array
19.5 An Industrial Example
19.6 Revision Questions
Further Reading
20 Laboratory 9: Wind Power (Electrical Components)
20.1 Introduction
20.2 Generator Choices
20.3 Physical Parameters for the Laboratory Wind Turbine
20.4 The Generator and Grid Side Architectures
20.5 Background Theory
20.6 Generator Side Model
20.7 Generator Side Control Law
20.8 The Link Capacitor Model
20.9 Regulation of the Capacitor Voltage
20.10 Model for the Grid Side Transformer
20.11 The Grid Side Control Law
20.12 Complete Electrical System Control Law
20.13 Testing the Electrical Control Laws
20.14 Experiments on the Complete System
20.15 Revision Questions
Further Reading
Note
21 Laboratory 10: Cross‐Directional Control in Paper Machines: PID Control
21.1 Introduction
21.2 Web‐Forming Process
21.3 Basis Weight Control in a Paper Machine
21.4 Process Model
21.5 Simple SISO Design Ignoring Coupling
21.6 Simple SISO Design Accounting for Coupling
21.7 Summary
21.8 Revision Questions
Further Reading
Notes
Part VI: Multivariable Control Systems (More General Methods)
22 State Variable Feedback
22.1 Introduction
22.2 Sampled‐Data Control
22.3 Dynamic Programming
22.4 Infinite Horizon Linear Quadratic Optimal Problem
22.5 Delta‐Domain Result
22.6 Continuous‐Time Linear Quadratic Regulator
22.7 Regulation to a Fixed Set‐Point
22.8 Frequency Domain Insights into the Linear Quadratic Regulator
22.9 Output Feedback
22.10 Separation
22.11 Achieving Integral Action
22.12 All Stabilising Control Laws Revisited
22.13 Model Predictive Control
22.14 Revision Questions
Further Reading
Notes
23 The Kalman Filter
23.1 Introduction
23.2 Periodic Disturbances
23.3 The Best Observer Gain
23.4 Steady‐State Optimal Estimator
23.5 Treating Non‐White Noise
23.6 Dealing with Constant Disturbances
23.7 Periodic Disturbances
23.8 Accounting for Delays
23.9 Multiple Output Measurements
23.10 Continuous‐Time Kalman Filter
23.11 Linking Continuous Kalman Filter and Discrete Kalman Filter
23.12 The Linear Quadratic Regulator Revisited
23.13 Quantifying the Performance
23.14 Revision Questions
Further Reading
Notes
24 Laboratory 11: Rolling Mill Revisited (Periodic Disturbances)
24.1 Introduction
24.2 Disturbances
24.3 Effects of Roll Eccentricity
24.4 Tight Feedback Control
24.5 Eccentricity Compensation
24.6 Optimal Observer Design
24.7 Eccentricity Compensation Using the Kalman Filtering
24.8 Conclusion
24.9 Revision Questions
Further Reading
Notes
Part VII: Introduction to the Modelling and Control of Nonlinear Systems
25 Modelling and Analysis of Simple Nonlinear Systems
25.1 Introduction
25.2 Errors Arising from Large Actuator Movement
25.3 Nonlinear Correction by Gain Change
25.4 Nonlinear Correction by Cascade Control
25.5 Saturation
25.6 Extension to Rate Limitations
25.7 Minimal Actuator Movement
25.8 Describing Function Analysis
25.9 Predicting the Period and Amplitude of Oscillations
25.10 Revision Questions
Further Reading
26 Laboratory 12: Continuous Casting Machine (Nonlinear Considerations)
26.1 Introduction
26.2 The Slide Gate Valve
26.3 Investigation of Effect of Nonlinear Valve Geometry
26.4 An Explanation for the Observed Oscillations
26.5 A Redesign to Account for Slip‐Stick Friction
26.6 Revision Questions
Further Reading
27 Laboratory 13: Cross‐Directional Control (Robustness and Impact of Actuator Saturation)
27.1 Introduction
27.2 Effect of Actuator Saturation Without Anti‐Windup Protection
27.3 PI Decoupled Design with Simple Anti‐Windup Protection
27.4 Conditioning Problems
27.5 PI Decoupled Design with Anti‐Windup Protection Limited to Low Spatial Frequencies
27.6 PI Decoupled Design with Adaptive Spatial Frequency Selection
27.7 Conclusions
27.8 Revision Questions
Further Reading
Note
Part VIII: Modelling and Control of More Complex Nonlinear Systems
28 Modelling of a Rocket in Three‐Dimensional Flight
28.1 Introduction
28.2 Preliminaries
28.3 Translational Dynamics
28.4 Rotational Dynamics
28.5 Stable or Unstable Rocket
28.6 Revision Questions
Further Reading
Note
29 Modelling of a Steam‐Generating Boiler
29.1 Introduction
29.2 Physical Principles
29.3 Physical Principles Used in Boiler Modelling
29.4 Mass Balances
29.5 Constant Volume of Drum, Risers and Downcomers
29.6 Energy Balances
29.7 A Model for Boiler Pressure
29.8 A Model for Drum Water Level
29.9 Spatial Discretisation and Homogeneous Mixing in the Risers
29.10 Water Flow in the Downcomers
29.11 Superheaters
29.12 Steam Receiver
29.13 Other Model Components
29.14 Revision Questions
Further Reading
Note
30 Laboratory 14: Control of a Steam Boiler
30.1 Introduction
30.2 Extracting an Approximate Linear Model
30.3 The Control Architecture
30.4 Regulating Steam Flow from the Boiler
30.5 Boiler Pressure Controller
30.6 Drum Water Level Controller
30.7 Steam Receiver Controller
30.8 Experiments
30.9 Summary
30.10 Revision Questions
Further Reading
Note
Index
End User License Agreement
List of Tables
Chapter 4
Table 4.1 Laplace transform table.
Table 4.2 Properties of Laplace transforms.
Chapter 7
Table 7.1 Ziegler–Nichols tuning, using the oscillation method.
Chapter 13
Table 13.1 Laboratory tests.
Table 13.2 Laboratory tests.
Table 13.3 Laboratory tests.
Chapter 17
Table 17.1 Operator and transform relationships.
Chapter 29
Table 29.1 List of variables.
List of Illustrations
Chapter 1
Figure 1.1 Screenshot of Continuous Caster Laboratory.
Chapter 2
Figure 2.1 Schematic of a simple single‐input single‐output system.
Figure 2.2 Schematic of simple feedback system.
Figure 2.3 Schematic of a simple single‐input single‐output system with meas...
Figure 2.4 Schematic applying feedback control to a simple single‐input sing...
Chapter 3
Figure 3.1 Radio telescope – Parkes, NSW, Australia.
Figure 3.2 Radio telescope positioning.
Figure 3.3 Band‐pass filter electrical circuit.
Figure 3.4 Inverted pendulum schematic.
Figure 3.5 Flow out of tank.
Figure 3.6 Second‐order modulator.
Chapter 4
Figure 4.1 Electric forced air‐heating system.
Figure 4.2 Discrete pulse.
Figure 4.3 Step‐response indicators.
Figure 4.4 Exact (blue) and asymptotic (red) Bode plots.
Chapter 5
Figure 5.1 The physical equipment.
Figure 5.2 Screen shot of program.
Figure 5.3 The servomechanism.
Figure 5.4 Block diagram of the servo system.
Chapter 6
Figure 6.1 Simple feedback control system.
Figure 6.2 Single‐zero function and Nyquist analysis. (a) inside region an...
Figure 6.3 Nyquist path.
Figure 6.4 Modified Nyquist path.
Figure 6.5 Stability margins and sensitivity peak. (a) Phase and gain margin...
Figure 6.6 Stability margins in Bode diagrams.
Figure 6.7 Nyquist plot for the nominal and the true loop.
Chapter 7
Figure 7.1 Steam receiver schematic.
Figure 7.2 Steam receiver root locus of closed‐loop poles where control law ...
Figure 7.3 Steam receiver root locus of closed‐loop where control law has on...
Figure 7.4 Locus for the closed‐loop poles when the controller zero varies....
Figure 7.5 Youla parameterisation of all stabilising controllers.
Figure 7.6 Youla parameterisation of all stabilising controllers with anti‐w...
Figure 7.7 Basic feedback control loop.
Figure 7.8 Bode magnitude plot of exact derivative (dashed) and filtered der...
Figure 7.9 Disturbance feedforward structure.
Figure 7.10 Question 3 RLC circuit.
Chapter 8
Figure 8.1 The servomechanism.
Figure 8.2 Block diagram.
Chapter 9
Figure 9.1 Continuous caster schematic.
Figure 9.2 Screenshot of continuous caster laboratory.
Figure 9.3 Overall view of the continuous caster at BHP, Newcastle, Australi...
Figure 9.4 The steel is pouring through the slide gate valve.
Figure 9.5 The continuously cast steel passing through the secondary cooling...
Figure 9.6 Simplified open‐loop model.
Figure 9.7 Closed loop.
Figure 9.8 Control loop with disturbance feedforward.
Chapter 10
Figure 10.1 Coupled tanks virtual lab interface.
Figure 10.2 Coupled tanks schematic.
Figure 10.3 Coupled tanks block diagram.
Figure 10.4 Coupled tanks control block diagram.
Chapter 11
Figure 11.1 Photo of wind turbines.
Figure 11.2 Screenshot of anemometer and wind speed.
Figure 11.3 Main screen.
Figure 11.4 Blade cross‐sectional view.
Figure 11.5 Block diagram of yaw control.
Figure 11.6 One over time constant versus gain.
Figure 11.7 Air flow for air turbine .
Figure 11.8 Tracking of maximum power depending on the wind speed.
Figure 11.9 Block diagram of rotational velocity control.
Figure 11.10 Saturation of rotational speed set point.
Figure 11.11 Power control regions.
Figure 11.12 Block diagram of pitch control.
Chapter 12
Figure 12.1 Youla–Kucera delay block diagram.
Figure 12.2 Smith predictor block diagram.
Figure 12.3 Impact of delay.
Figure 12.4 Bode plot of real‐world system.
Chapter 13
Figure 13.1 Single stand 4 high cold rolling mill.
Figure 13.2 Typical measurements and control signals in a single stand.
Figure 13.3 Schematic of rolling mill laboratory.
Figure 13.4 Open‐loop rolling mill model.
Figure 13.5 Single degree‐of‐freedom control loop using measured exit thickn...
Figure 13.6 Smith predictor architecture.
Chapter 14
Figure 14.1 Approximate Youla–Kucera parameterisation for unstable plant.
Chapter 15
Figure 15.1 Screen shot.
Figure 15.2 2D rocket model ( is positive if the centre of pressure is abov...
Figure 15.3 Nyquist plot.
Chapter 16
Figure 16.1 Simple continuous‐time feedback circuit.
Figure 16.2 Graphical interpretation of the Bode integral.
Chapter 17
Figure 17.1 A/D converter representation including anti‐aliasing filter.
Figure 17.2 D/A converter using zero order hold. (a) input and (b) output.
Figure 17.3 Stability domain for shift operator models.
Figure 17.4 Stability domain for delta operator models.
Figure 17.5 Question 10 diagram.
Chapter 18
Figure 18.1 Screenshot of program.
Figure 18.2 Developing a filter.
Figure 18.3 Nearest neighbour 2‐bit quantisation.
Figure 18.4 Feedback quantiser.
Figure 18.5 Noise shaping quantiser as a two‐degree‐of‐freedom feedback syst...
Figure 18.6 The quantiser as a noise source.
Figure 18.7 Feedback loop.
Chapter 19
Figure 19.1 Heating problem.
Figure 19.2 Original control actuator and plant.
Figure 19.3 Added inner control loop.
Figure 19.4 Cascade architecture.
Figure 19.5 Response to disturbance.
Figure 19.6 Response to disturbance with disturbance feedback.
Figure 19.7 Simplified boiler architecture.
Chapter 20
Figure 20.1 Wind turbine lab screenshot showing the generator side controlle...
Figure 20.2 Block diagram of electrical system.
Figure 20.3 Voltage and current vectors in frame.
Figure 20.4 Grid side control.
Figure 20.5 The full electrical control system. (a) Electrical torque loop, ...
Figure 20.6 Distribution graph of wind direction – wind rose.
Figure 20.7 Phase locked loop.
Chapter 21
Figure 21.1 Screenshot of program.
Figure 21.2 Generic web forming process.
Figure 21.3 CD profile for a unit step movement in actuator No. 6: generic w...
Chapter 23
Figure 23.1 Radar range and bearing.
Chapter 24
Figure 24.1 Screenshot of program showing the observer control system.
Figure 24.2 Roll eccentricity exaggerated for illustration.
Figure 24.3 Feedback system.
Chapter 25
Figure 25.1 Real valve flow dynamics.
Figure 25.2 Cascaded loops.
Figure 25.3 Feedback system.
Figure 25.4 Feedback system with saturation.
Figure 25.5 Saturation‐aware observer.
Figure 25.6 Relay function.
Figure 25.7 Output of relay to a sinusoidal input.
Figure 25.8 Closed‐loop with relay.
Figure 25.9 Question 3 block diagram.
Figure 25.10 Relay function with dead zone.
Chapter 26
Figure 26.1 Screenshot of continuous Caster laboratory.
Figure 26.2 Control loop.
Figure 26.3 Slide gate valve.
Figure 26.4 Model of the valve with nonlinear characteristics and additional...
Figure 26.5 Valve with slip‐stick friction.
Figure 26.6 The relationship between the input and output for the nonlinear ...
Figure 26.7 Valve with slip stick friction in mould level controller.
Figure 26.8 The schematic of the nonlinear block.
Figure 26.9 Slip stick friction compensated with dither.
Figure 26.10 Backlash nonlinearity.
Chapter 27
Figure 27.1 Anti‐windup controllers in diagonal space. (a) General anti‐wind...
Chapter 28
Figure 28.1 Rigid body in free space.
Chapter 29
Figure 29.1 Cross‐section of boiler.
Figure 29.2 Simulation results.
Figure 29.3 Continuously stirred tank reactor schematic.
Chapter 30
Figure 30.1 Boiler laboratory drum water and pressure controller.
Figure 30.2 Boiler laboratory drum water, drum pressure and steam receiver c...
Figure 30.3 Electrical analogue of a furnace.
Guide
Cover
Table of Contents
Title Page
Copyright
Preface
About the Authors
Acknowledgements
About the Companion Website
Begin Reading
Index
End User License Agreement
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Practical Control System Design
Real World Designs Implemented on Emulated Industrial Systems
Adrian MedioliUniversity of NewcastleAustralia
Graham GoodwinUniversity of NewcastleAustralia
This edition first published 2024© 2024 John Wiley & Sons Ltd
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Preface
This book provides an introduction to control system design. It would be suitable as the basis of a first or second course in control. It has also been written as a refresher course for graduate engineers who wish to ‘up‐skill’ in the area of practical control system design.
The book has been inspired and informed by the authors' industrial experience in control. Our goal has been to package that experience so that those reading this book and using the associated resources will acquire the tools and skills necessary to tackle real‐world control engineering design problems.
The book draws upon many industrial projects conducted by the authors and associates. These projects are used as case studies throughout the book. The authors, belief is that the richest learning experience comes from testing ideas via ‘hands‐on’ laboratory studies that closely reflect the real world. The case studies are organised in the form of virtual laboratories. Readers can explore the ideas at their own pace and to their own level of interest.
The case studies include the following:
electromechanical servo systems
fluid storage
continuous steel casting
rolling mill centre line gauge control
rocket dynamics and control
cross‐directional control in paper machines
audio quantisation
wind power generation – mechanical aspects
wind power generation – electrical aspects (including three‐phase induction generator)
boiler control.
Wherever possible, the models used in the case studies use physical dimensions and parameters from real‐world systems.
The laboratories can be conducted ‘on‐line’ without needing to access a physical laboratory. Thus, the laboratories would be well suited to remote education. Nonetheless, the laboratories have been designed so as to be as realistic as possible. Indeed sitting in front of the laboratory interface is essentially the same as sitting in an industrial control room with the following two caveats:
It would be completely unrealistic for any one person to gain access to the range of physical systems studied here.
One can learn by one's mistakes on a virtual laboratory whereas, if such mistakes were ever made in practice, the consequences could be catastrophic.
For ease of use, the architectures of the various control laws have been provided. Readers can use these architectures as a starting point. In many cases, it is also possible to link the process simulations to a control law whose architecture is specified by the reader or by an instructor.1 In either case, it is suggested that the reader begin with the given architectures to establish a starting point for further study.
The topics covered in the course include:
modelling
PID design
dealing with delays
impact of actuator limitations
dealing with sensor limitations
feedforward
soft sensing
dealing with nasty disturbances
observers and Kalman filtering
Bode sensitivity constraints
dealing with periodic disturbances
multivariable interactions
sampled data systems
dealing with resonant behaviour
controlling systems which are open‐loop unstable.
This book contains 30 chapters. Sixteen of these chapters provide a review of the necessary background theory, whilst the remaining 14 are devoted to testing the ideas on real‐world (emulated) systems. The treatment of theoretical topics will be relatively brief since the key aim of the book is to study real‐world control system design.
The authors' belief is that anybody who has understood the core concepts explained in the book will be well‐equipped to tackle real‐world control design problems.
Difficult choices need to be made when writing a book of this type. In particular, to present a detailed treatment of any of the topics covered in the theory sections would expand the book enormously. Hence, the driving philosophy has been to limit the presentation to the depth necessary to enable the reader to undertake the real‐world designs. In other words, the goal has been to make the presentation ‘as simple as possible, but not so simple as to lose important details’.
Note
1
Exceptions are the Wind Power Laboratory and the Audio Laboratory which require specific architectures.
About the Authors
Adrian Medioli, BE, PhD, OStJ, MIEEE
Adrian Medioli was born in Newcastle, Australia, and completed a B.E. (Computer) in 1992 at the University of Newcastle. He spent 10 years as a senior systems engineer working on the automation and control of steel manufacturing processes including rolling mills, galvanising lines, batch annealing, gauge control and Sendzimir mills and two years as a director of Omni Automation Pty. Ltd., a control engineering and automation startup company. In 2008, he completed a Ph.D. in Electrical Engineering at the University of Newcastle. From 2008 to 2021, he was employed as a research academic in the ARC/PRC for Complex Dynamic Systems and Control at the University of Newcastle. In 2021, he joined Whitely Corporation where he worked on the implementation of Industry Four protocols. As a professional engineer Adrian has worked on many and varied projects which resulted in extensive experience in the areas of software development, team management, control system design and implementation to steel production and manufacturing. As a researcher, he has developed and implemented new techniques in optimisation and control. Some of his achievements include a new technique for maximal controllability of unstable systems using reduced complexity Model Predictive Control (MPC); development of tools for the classification of coal loader downtime causes; ambulance fluid deployment strategy optimisation; ambulance optimal distribution and rostering to better match demand; design and testing of novel control strategies with application to the development of an artificial pancreas for better blood glucose regulation in diabetes sufferers; and re‐design and development of a suite of virtual laboratories for the teaching of control system design. At present, Dr. Medioli's focus is on his research interests, including optimisation‐based control strategies such as MPC stabilisation and modelling with particular emphasis on unstable systems; virtual laboratory development; and the application of control concepts and design to medical problems. He is a Member of IEEE and holds a Certificate IV in vocational training and assessment with accreditation under the Australian Quality Training Framework (AQTF).
Graham Goodwin, BSc, BE, PhD, FRS, FIEEE, Hon.FIE.Aust., FTSE, FAA
Graham Goodwin is an Emeritus Laureate Professor of Electrical Engineering at the University of Newcastle. His education includes B.Sc., B.E., and Ph.D. from the University of New South Wales. In 2010, he was awarded the IEEE Control Systems Field Award, in 2011 the ACA Wook Hyuan Kwon Education Award and in 2013 he received the Rufus T. Oldenburger Medal from the American Society of Mechanical Engineers. He was twice awarded the International Federation of Automatic Control triennial Best Engineering Text Book Prize. In 2021, he was awarded the American Control Council John Ragazzini Education Award. He is a Fellow of IEEE; an Honorary Fellow of Institute of Engineers, Australia; a Fellow of the International Federation of Automatic Control; a Fellow of the Australian Academy of Science; a Fellow of the Australian Academy of Technology, Science and Engineering; a Member of the International Statistical Institute; a Fellow of the Royal Society, London, and a Foreign Member of the Royal Swedish Academy of Sciences. In 2021, he was recognised by the Australian Government by becoming an Officer in the General Division of the Order of Australia. He holds Honorary Doctorates from Lund Institute of Technology, Sweden, and the Technion Israel. He is the co‐author of 10 previous books, four edited books, and five hundred papers. He holds 16 International Patents covering rolling mill technology, telecommunications, mine planning, and mineral exploration. His current research interests include power electronics, boiler control systems and management of type 1 diabetes.
Acknowledgements
This book is the culmination of many years of work by many individuals. An abbreviated list of those who worked on developing the laboratories includes Adrian Basitani, Diego Carrasco, Ross Cockwell, Sam Crisafulli, David Ferris, Arie Feuer, Richard Middleton, Galina Mirzaeva, Rob Peirce, Osvaldo Rojas, Frank Sabora, Maria Seron, Eduardo Silva, Bob Skelton, Peter Stepien, James Welsh, Peter Wellstead, and Mark West. Financial support from the Australian Research Council plus the School of Engineering and the College of Engineering, Science and Environment at the University of Newcastle is gratefully acknowledged. The book was typed by Jayne Disney.
Adrian Medioli
Graham Goodwin
About the Companion Website
This book is accompanied by a companion website.
www.wiley.com/go/medioli/practicalcontrolsystemdesign
This website includes:
Solutions
Software
Solutions to revision questions
Part IModelling and Analysis of Linear Systems
This part begins the journey through the fascinating world of control system design. Chapter 1 provides a brief historical overview of control. Chapter 2 explains why control is a quintessential example of an inverse problem. Chapter 3 provides an introduction to modelling of physical systems. Chapter 4 describes basic mathematical tools used to analyse linear systems. Chapter 5 contains the first virtual laboratory. A simple door‐closing mechanism is used to illustrate basic principles of model building using data collected from a system. Many of the ideas covered in this part of the book would normally be presented in an introductory course on control. Hence, the development will focus on the key concepts that will be used in the real‐world designs covered in the sequel.
1Introduction to Control System Design
1.1 Introduction
Control is the hidden yet ubiquitous technology [1]. Essentially, no piece of equipment in the modern world would operate satisfactorily without some form of feedback control. Indeed, beyond physical devices such as automobiles or aircraft, control lies at the core of many other systems, e.g. economic systems and even human psychology.
A fascinating overview of control is contained in reference [2] given in Further Reading for this chapter.
The essence of feedback control is that one wishes to act on a system so that it performs in some desirable fashion. Often the goals are expressed as ‘set‐points’ for the outputs, sometimes called ‘Process Variables’ (PVs). Action on the system is achieved by changing the inputs, sometimes called ‘Manipulated Variables’ (MVs). Examples occur in every realm. Just a few examples to think about are:
making a small company profitable
ensuring a national economy has a low inflation rate
delivering electricity to a community or country
cooling a house
landing an aircraft
producing a desired product in a chemical plant
achieving a personal fitness goal
designing an adaptive cruise controller for a car
achieving a certain level of proficiency in a trade or discipline
taking a drug so as to control a chronic disease.
For each of the above examples, the reader may like to think about what the PVs and MVs could be.
The tasks listed above would be easy if
one knew the exact impact that changing the MVs have on the PVs (i.e. one has an exact model), and
there were no external disturbances.
Alas, in practice, neither of the above requirements is met. Thus, the art of control system design is to build a ‘control’ system that achieves the desired goals, as closely as possible, in the face of uncertainty in both the model and external disturbances.
1.2 A Brief History of Control
Control underpins the operation of almost every piece of modern equipment, including automobiles, aircraft, chemical plants, national economies and medical devices. It is thus not surprising that people have been thinking about control for hundreds of years. Indeed, Ref.[3] describes a feedback mechanism aimed at regulating the accuracy of a water clock. This device was developed several thousand years ago.
A very well‐known application of feedback control was the regulation of the speed of steam engines in the eighteenth‐century CE. This was the work of Watt and colleagues and was an enabling technology in the industrial revolution. These speed governors worked amazingly well but were known to sometimes exhibit oscillatory behaviour. James Clarke Maxwell, who is famous for his work on electromagnetic theory, provided a theoretical description in (1868) for this in terms of the properties of ordinary differential equations in the time domain [4].
An important early practical discovery was the use of ‘integral action’ to achieve zero steady‐state error. This was introduced in 1790 in a governor designed by the Perrier brothers [2].
Turning to the process industries, control is an essential technology in all chemical processes. Indeed, modern chemical plants are ‘littered’ with controllers. The most common controller is a device known as a PID controller. The letters PID stand for Proportional, Integral and Derivative feedback. Though simple, such controllers are incredibly robust and achieve remarkable performance in many cases.
Another remarkable area where control is essential is in powered flight. Indeed, the Wright Brothers, aircraft critically depended on the pilot adjusting the wing surfaces to maintain stability of flight [5].
Moving to an entirely different area, feedback control is a key enabling technology in the telephone. The telephone was invented by Alexander Graham Bell in 1876. However, as the number of repeaters was increased internally generated noise and distortion became intolerable. A major breakthrough was made by Harold Black at Bell Laboratories in 1927 with the development of the negative‐feedback amplifier [6]. The essential idea was to use feedback around a high‐gain amplifier to reduce the impact of noise on the output signal. In these feedback amplifiers, instability was again sometimes observed (appearing as a ‘whistle’). A remarkable engineer, Harry Nyquist, studied the problem of stability at Bell Laboratories [7]. He moved away from the time domain and instead studied how sinusoidal signals propagated around a feedback loop. This was the initial step in using frequency domain analysis of feedback systems.
The frequency domain insights were a huge breakthrough. This was built upon by many people. For example, major contributions were made by Bode in 1940. He recognised that complex variable theory could be used to give deep insights into the frequency domain analysis of feedback systems [8]. In a truly remarkable result, he proved that the integral of log sensitivity with respect to frequency was constant. (Actually zero for an open‐loop stable system.) This meant that the action of feedback was simply to shift the impact of disturbances around in the frequency domain. In particular, reducing sensitivity to disturbances at low frequencies would necessarily be accompanied by an increase in sensitivity to disturbances at higher frequencies. This idea underpins many systems used in audio quantisation, industrial electronics and process control.
The above body of work, which emphasises frequency domain concepts, is often classified as part of the ‘Classical Control’ era. This circle of ideas dominated until the 1960s when a step back to time domain ideas was made. A major contribution was the work of Pontryagin, Boltyanskii, Gamkrelidze and Mishechenko (1962) in the USSR on ‘optimal’ control [9]. A related stream of research was the development of Dynamic Programming by Richard Bellman in the United States [10]. The linear quadratic version of these ideas gave a particularly elegant solution as shown by Bellman, Kalman, Bucy and many others [11]. A dual problem turns out to be that of estimating the ‘state’ of a system from output measurements. Algorithms which carry out this process are commonly called ‘Observers’. Major contributions to the subject of observers were made by Stratonovich, Luenberger, Kalman, Bucy, Wonham and many others [12–14].
The latter ideas are sometimes described as being part of the ‘Modern Control’ era. In the late 1980s a return was made again to frequency domain ideas. This included work on robust linear control by Doyle, Glover, Khargonekar and Francis [15] plus many others. It was recognised that these new, time domain, results gave elegant solutions. Not surprisingly, the combination of time domain and frequency domain gives the most powerful view. Thus, the modern view of control involves a mixing of both time and frequency domain ideas.
1.3 Digital Control
Early control laws were implemented in analogue form using hydraulic or pneumatic equipment. An important development that occurred in control was the introduction of digital feedback using computers. Early work emphasised the difference between analogue control and digital control. This led to a new set of challenges in control system design. For example, linear analogue control loops are known to be stable if the roots of the characteristic polynomial lie in the open‐loop left‐half complex plane, whereas digital control loops are stable if the roots of the characteristic polynomial (expressed in terms of the forward shift operator) lie in the open unit disc. An entirely new machinery was developed to study such problems including shift operators and Z‐transforms [16].
As with the time‐frequency domain divide, the greatest insights arise when both continuous and discrete viewpoints are combined. In support of this perspective, it has been shown that analogue and digital control loops can be studied under a common framework using the divided difference operator [17]. Using this operator, analogue control can be viewed as a special case of digital control in which the sample period is made arbitrarily small. This line of work has a long history in mathematics, but its relevance to control engineering was brought to the notice of control engineers in a series of papers by one of the authors of the current book (Goodwin) together with co‐workers Middleton, Poor, Feuer, Yuz and many others [18].
1.4 Our Selection
The brief history given above gives just a preliminary ‘taste of’ the huge range of ideas that exist in modern control engineering.
Of course, there are many other ideas and concepts in modern control science. For example, the interested reader is referred to the excellent survey by Åström and Kumar [2]. Other topics of great importance include Adaptive Control, Nonlinear Control, Control of Discrete Event Systems, Model Predictive Control and many others.
It would be impossible in a single book to do justice to even a small fraction of this huge spectrum of ideas. Thus, the focus of the book will be a relatively small subset. The choice is driven by the author's experience in using control to solve real‐world control problems in industry. This leads to a particularly simple path.
1.5 Thinking Outside the Box
A recurring theme in the real‐world control problems described in this book is that one often needs to ‘think outside the box’. Indeed, it is a belief of the authors that, on occasions, excessive emphasis is devoted to the tuning of control law parameters within a given control architecture rather than focusing on the architecture itself.
Undoubtedly, getting the right tuning is important and thus significant space will be devoted to this topic in this book. However, the biggest improvements in control performance often arise from changing the underlying architecture of the system. This can be achieved either by introducing new sensors, by changing the physical structure of the system or by introducing alternative or secondary actuators. As a simple example, pure delays in measured outputs fundamentally limit the achievable performance. (Imagine trying to steer a car with your eyes closed and somebody telling you where the car sits on the road but with a 10‐seconds delay!) In some cases, the problem can be remedied by introducing a new sensor. A very nice example of this will be given in Chapter 13. There, a device known as a BISRA gauge will be studied. This device is used for centre line thickness control in rolling mills. Other examples of the benefits of using alternative sensors or actuators will arise in other real‐world problems.
1.6 How the Book Is Organised
Chapters 2–4 contain basic modelling and mathematical background necessary to provide a common ‘language’ that can be used to discuss feedback control problems. It is not surprising that control has its own ‘language’. After all, one would not want to keep having to explain what ‘PID’ means. Instead, it is better to simply view this as part of the control lexicon. From Chapter 5 onwards, the presentation will ‘shift gears’ and control ideas and language will be utilised in the context of real‐world problems of the type illustrated in Figure 1.1. Extensive use will be made of the Virtual Laboratories to motivate and illustrate the evolving circle of ideas. The sequence of laboratories has been organised so that the ideas evolve in a ‘structured’ fashion proceeding from relatively simple ideas to more complex concepts. New concepts and methods are introduced, if and only if, they are needed for the real‐world problems. The reader may feel a little frustrated that Chapters 1–4 are largely of a theoretical nature. However, some basic background is necessary before one can jump into a practical problem. Readers who have background in basic control concepts could go straight to Chapter 5.
Figure 1.1 Screenshot of Continuous Caster Laboratory.
The book has been divided into a number of parts. The reason for doing this is to mark the transition in ideas from simple to more complex.
1.7 Testing the Reader's Understanding
Throughout the book, the reader is asked to perform various experiments and to answer some revision questions. At times, these questions are deliberately ‘open‐ended’ so as to challenge the reader to think outside the bounds of the material presented in the book.
1.8 Revision Questions
List five devices in your possession which depend upon feedback for their operation.
The term ‘feedback’ is often used when somebody provides a commentary on somebody's work. What are the measurements and manipulated variables?
Use your answer to Question 2 to illustrate why not all feedback is helpful.
Further Reading
1
K.J. Åström, “Automatic control ‐ the hidden technology”, in P. Frend (ed.)
Advances in Control ‐ Highlights of ECC'99
(pp. 1–29), European Control Council, Springer–Verlag.
2
K.J. Åström and P.R. Kumar, “Control a perspective”,
Automatica
, 50:3–43, 2014.
3
O. Mayr,
The origins of feedback control
, MIT Press, 1970.
4
J.C. Maxwell, “On governors”,
Proceedings of the Royal Society of London
, 16:270–283, 1868.
5
C.S. Draper, “Flight control”,
Journal Royal Aeronautical Society
, 59:449–477, 1955.
6
H.S. Black, “Stabilizing feedback amplifies”,
Bell System Technical Journal
, 13:1–18, 1934.
7
H. Nyquist, “Regeneration theory”,
Bell System Technical Journal
, 11:126–147, 1932.
8
H. Bode,
Network analysis and feedback amplifier design
, Van Nostrand, New York, 1945.
9
L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishechenko,
Mathematical theory of optimal processes
, John Wiley & Sons, Inc., New York, 1962.
10
R.E. Bellman,
Dynamic programming
, Princeton University Press, Princeton, New Jersey, 1957.
11
R.E. Kalman,“Contributions to the theory of optimal control”,
Boletin de la Sociedad Matematica Mexicana
, 5:102–119, 1956.
12
R.E. Kalman and R.S. Bucy, “New results in linear filtering and prediction theory”,
Transactions of the ASME, Series D, Journal of Basic Engineering
, 83(1):95–108, 1961.
13
D.G. Luenberger,“Observing the state of a linear system”,
IEEE Transactions on Military Electronics
, 8(2):74–80, 1964.
14
W. Wonham,“Dynamic observers ‐ geometric theory”,
IEEE Transactions on Automatic Control
, 15(2):258–259, 1970.
15
J. Doyle, K. Glover, P. Khargonedor and B. Francis, “State‐ space solutions to standard and control problems”,
IEEE Transactions on Automatic Control
, 34(8):831–847, 1989.
16
J.R. Ragazzini and L.A. Zadeh, “The analysis of sampled‐data systems”,
Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry
, 71(5):225–234, 1952.
17
G.C. Goodwin, R.H. Middleton and H.V. Poor, “High‐speed digital signal processing and control,
Proceedings of the IEEE
, 80:240–259, 1992.
18
G.C. Goodwin, J.C. Agüero, M.E.C. Garridos, M.E. Salgado and J.I. Yuz, “Sampling and sampled‐data models. The interface between the continuous world and digital algorithms”,
IEEE Control Systems Magazine
, 33(5):34–53, 2013.
2Control as an Inverse Problem
2.1 Introduction
This chapter takes a first look at control system design. The presentation begins by viewing control as a simple example of an inverse problem. Indeed, inversion lies at the core of all control system design procedures and is thus a unifying theme.
2.2 The Elements
The most elementary control problem is shown in Figure 2.1.
With reference to Figure 2.1, say that one wishes to act on the process, , via the input , so as to cause the output, , to reach (as closely as possible) a given ‘set‐point’, , in the presence of disturbances, .
At a conceptual level, the system of Figure 2.1 can be described as follows:
For the moment, the reader can assume that are all simple scalar real numbers.
Say that it is desired that, be equal to , then it is clear from (2.1) that the following choice for the input will ‘do the trick’.
From Eq. 2.2 we note that it contains an inverse and it is evident that the following information is needed:
knowledge of ,
measurement of ,
and the ability to sensibly invert .
The astute reader will immediately raise serious objections to the model (2.1) and point out that the real‐world problems described in the introduction are never so simple that they could possibly be described by a simple algebraic gain. This is, of course, true. However, remarkably, it will be shown later that a model consisting of a simple algebraic gain covers a huge range of real cases provided one uses a simple trick of thinking about the model in a particular way. For those readers familiar with eigenvalues and eigenvectors from linear algebra, the trick is to use this idea. Other readers may be familiar with frequency response which, again, uses the same concept.
Figure 2.1 Schematic of a simple single‐input single‐output system.
2.3 Using Eigenvalue Analysis
To illustrate the idea of how complicated systems can be represented by simpler (scalar) systems, a brief diversion is taken to use the idea of eigenvalues to simplify a multi‐input, multi‐output system. Readers who are not familiar with eigenvalues need not read the remainder of this section but can immediately jump to Section 2.4.
Consider a system described by a linear (non‐dynamic) matrix operation
where is an output vector and is an input vector, is an matrix of reals.
Say that one can diagonalise the Matrix operator by a transformation of the form:
where is a matrix ‘diagonaliser’ and
where are the eigenvalues. Substituting (2.4) into (2.3) yields
or
Hence, instead of using the original input vector and the original output vector , one might use instead
Then, the model becomes
However, this is nothing more than a set of scalar gain systems, i.e.
The book will return to this idea frequently. Indeed, this is the core principle underlying Laplace transforms and frequency domain analysis. Specifically, the Laplace operator turns out to be the eigenvalue of systems described by linear differential equations. For the moment, it suffices to consider the simple scalar algebraic model of Eq. 2.1. However, the reader will be encouraged to know that these simple algebraic ideas will extend rather easily to systems described by high‐order linear differential equations, as shown later in Chapter 4. Later in the book, specifically in Section 21.6, an industrial application of the above idea will be described that directly uses the core idea given in Eq. 2.11.
2.4 The Effect of Process and Disturbance Errors
Another criticism of the solution (2.2) is that it ignores errors in the knowledge available about both the process and of the disturbance .
To illustrate this difficulty, say that it is believed that the model is and the disturbance is , however, in reality, the real plant and real disturbance satisfy
where and are errors.
Equation 2.1 then becomes
Alas, the control law (2.2) can only utilise what is believed to be true, i.e. it is only feasible to set
Substituting (2.15) into (2.14) yields
It can then be seen that what is actually achieved is
Then
or
The term is termed the ‘relative tracking error’. It follows from (2.19) that the relative tracking error is proportional to the relative error in one's knowledge of the model, i.e. , and the relative size of the error in the disturbance, i.e. .
2.5 Feedback Control
The state of affairs described in Section 2.4 is rather unsatisfactory since relative errors in and are reflected, undiminished, in the output response. Fortunately, there is a rather simple way that one can reduce the impact of the relative errors and on the output. The key idea is to use the ‘magic of feedback’, i.e. to replace the ‘open‐loop’ control system shown in Figure 2.1 by a ‘closed‐loop’ control system.
The basic idea inherent in feedback is to change the problem formulation via the use of feedback. The core idea is to compare the output, i.e. , to what is desired, i.e. and then to use the error to adjust via a feedback ‘gain’ .
This leads to a feedback (or closed‐loop) structure of the type conceptually illustrated in Figure 2.2.
Figure 2.2 Schematic of simple feedback system.
From Figure 2.2, the following algebraic relationships can be seen to hold. (Here it is again assumed that all quantities are simply scalar real numbers:)
Hence,
or
Examining the two terms on the right‐hand side of (2.23) leads to a rather magical conclusion, namely, if one could only make the gain very large then, independent of or , the following result would be achieved.
It thus follows that if one uses high gain feedback (i.e. large), then this will cause to equal independent of knowledge of or .
In case the reader should feel that this is of esoteric theoretical interest only, it will be shown in Section 4.10 that almost every practical control system uses (a special form of) infinite gain to achieve perfect tracking and disturbance rejection (for constant signals) without requiring knowledge of the true model of the system. This goes under the heading of ‘integral action’.
2.6 The Effect of Measurement Noise
Another type of error that is frequently encountered in real systems is measurement error. Indeed, the nature of sensors is such that the information they collect and communicate is usually corrupted by measurement errors and noise. This means that the measurements will have ‘noise’ superimposed on the information that is measured. Also, sensors can deliver inaccurate results for many reasons, including the inherent difficulty of measuring certain variables. These errors can be lumped into a measurement ‘noise’ term, .
Figure 2.3 Schematic of a simple single‐input single‐output system with measurement noise.
Figure 2.3 shows a simple process schematic, but now with measurement noise, , shown explicitly. Note that the output that is measured is described by the following equation:
where is the measurement noise and is the measured noisy output of the system.
The effect of noise on the simple feedback system can now be analysed.
A feedback system with both a disturbance, , and measurement noise, , is shown in Figure 2.4.
Figure 2.4 Schematic applying feedback control to a simple single‐input single‐output system with measurement noise.
From Figure 2.4, the following algebraic relationships follow:
Hence,
or
Note that now, instead of the conclusion fromSection 2.5, namely, that high gain feedback is sufficient to cause , now it is not true, in general, that due to the presence of measurement noise. Specifically making large ‘impresses’ the measurement noise, , on the output response.
2.7 Sensitivity Functions
Examining Eq. 2.30 shows that the right‐hand side depends on three key functions of the underlying quantities and . These functions play a key role in feedback control and they are given special names and symbols as follows:
Using these functions, then (2.30) can be rewritten as
Equation 2.34 suggests that and play a core role in feedback control problems. Indeed, understanding and will turn out to be one of the central themes as the ideas in the book evolve.
2.8 Reducing the Impact of Disturbances and Model Error
It can be seen from (2.32) and (2.33), that simply making large, causes and . This is great news since we see from the first two terms on the right‐hand side of (2.34) that, in the absence of noise, , the result follows.
2.9 Impact of Measurement Noise
The third term in (2.34) shows a fundamental difficulty, namely, making large, then . This leads to the conclusion that the measurement noise is ‘impressed’ on the output, i.e. , becomes 100% sensitive to measurement noise.
This leads to a fundamental design trade‐off in feedback design, namely,
‘One cannot simultaneously make the output of a closed‐loop system insensitive to both disturbances and measurement noise.’
This idea will also be a recurring theme as the book evolves.
2.10 Other Useful Sensitivity Functions
In addition to the sensitivity functions and , one can also define
It is easily checked that and are, respectively, the functions relating an input disturbance to the output and the set point to the input. The four sensitivity functions are sometimes referred to as the ‘gang of four’. They capture all possible combinations of and in a closedloop.
Whilst all four sensitivity functions have a role to play in control system design, it often suffices to focus on and . Thus, emphasis will be given to these two sensitivity functions in the sequel.
